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A333161
Triangle read by rows: T(n,k) is the number of k-regular graphs on n unlabeled nodes with half-edges.
7
1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 3, 4, 4, 3, 1, 1, 4, 8, 12, 8, 4, 1, 1, 4, 10, 24, 24, 10, 4, 1, 1, 5, 17, 70, 118, 70, 17, 5, 1, 1, 5, 24, 172, 634, 634, 172, 24, 5, 1, 1, 6, 36, 525, 4428, 9638, 4428, 525, 36, 6, 1, 1, 6, 50, 1530, 35500, 187990, 187990, 35500, 1530, 50, 6, 1
OFFSET
0,5
COMMENTS
A half-edge is like a loop except it only adds 1 to the degree of its vertex.
T(n,k) is the number of non-isomorphic n X n symmetric binary matrices with k ones in every row and column and isomorphism being up to simultaneous permutation of rows and columns. The case that allows independent permutations of rows and columns is covered by A333159.
T(n,k) is the number of simple graphs on n unlabeled vertices with every vertex degree being either k or k-1.
LINKS
FORMULA
T(n,k) = T(n, n-k).
EXAMPLE
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 2, 2, 1;
1, 3, 3, 3, 1;
1, 3, 4, 4, 3, 1;
1, 4, 8, 12, 8, 4, 1;
1, 4, 10, 24, 24, 10, 4, 1;
1, 5, 17, 70, 118, 70, 17, 5, 1;
1, 5, 24, 172, 634, 634, 172, 24, 5, 1;
1, 6, 36, 525, 4428, 9638, 4428, 525, 36, 6, 1;
...
The a(2,1) = 2 adjacency matrices are:
[0 1] [1 0]
[1 0] [0 1]
.
The A(4,2) = 3 adjacency matrices are:
[0 0 1 1] [1 1 0 0] [1 1 0 0]
[0 0 1 1] [1 1 0 0] [1 0 1 0]
[1 1 0 0] [0 0 1 1] [0 1 0 1]
[1 1 0 0] [0 0 1 1] [0 0 1 1]
CROSSREFS
Columns k=0..3 are A000012, A004526(n+2), A186417, A333163.
Row sums are A333162.
Central coefficients are A333166.
Sequence in context: A103691 A103441 A081206 * A156044 A180980 A275298
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Mar 11 2020
STATUS
approved